Do a full render for what I am seeing without changing the default camera?
Press Numpad 0 to view what the camera is seeing, select the new camera and press G Z Z (Which means move the camera along its own Z axis), move the mouse until you get the desired framing (like if you put a real camera closer or far from the subject) and confirm with left click.

You will get the same framing of the viewport only if the viewport and the camera have the same focal lenght (both are 50 mm as default).